You know the sample mean  of n observations. Once you know (n - 1) of the observations, show that you can find the remaining one. In other words, for a given value of , the values of (n - 1) observations determine the remaining one. In summarizing scores on a quantitative variable, having (n - 1) degrees of freedom means that only that many observations are independent. (If you have trouble with this, try to show it for n = 2, for instance showing that if you know that  = 80 and you know that one observation is 90, then you can figure out the other observation. The df value also refers to the divisor in s2 = ∑(x - )2/(n - 1).)